The definition of a frame function used in Section 
is based on the idea that every sequence of effects \(e_1,e_2,\ldots \in \mathcal {E}\) 
satisfying Eqs. (REF ) and (REF ) corresponds to an
observable. In quantum theory, however, a Gleason-type theorem can
already be derived by involving only a specific subset of all
POMs [1]} known as projective-simulable observables
[2]}. We will show now that a similar weakening of
the assumptions continues to imply the result of Theorem REF 
in the context of GPTs.
